Ask a trans advocate what a woman is. The only allowable definition is "anyone who identifies as a woman". That's not "less rigid", that's non-existent.
Do you think you might want to let them speak for themselves? You've got a very poor track record of accurately reporting what your opponents believe, or listening to what they say when asked what they believe.
The way notions of identity work are by distinguishing yourself from other people. It is some combination of what makes you like other people, e.g. "I can identify with that person", and what makes you stand out, e.g. "we identified the suspect out of a group of people". To identify as something, that thing has to have a meaning, not only for you but for other people. Identities are inherently relative to other people. If the definition of your identity is so vague as to be totally incomparable to the experience of others, it can't give structure or meaning to your life.
Identities don't solely or necessarily exist to "give structure or meaning to your life". They can also exist as simple descriptors. And identifying oneself in a certain way to others isn't always about structure or meaning either; it can be pure utility.
And then to completely structure your life around something like that, to build this identity fully into every social interaction, without ever being able to properly explain why, is to build on a foundation of nothing.
Trans people do this no more than cis people. You're just exaggerating it's focus for one group to give a false impression of the unreasonable.
Even an explanation as simple as "it's just what I like" would be sufficient to make some human connections, but an undefinable unexplainable feeling of what you are is just empty. It's a void of meaning. It tells me nothing about you, it tells you nothing about me, it tells you nothing about yourself.
You're the one stripping all meaning from the term, here, and then decrying how the term has become meaningless.
For instance, I can tell you my gender is male. Ok? I'm a "trans advocate", if you want to call it that.
Male gender has meaning and associations. Most trans people will agree-- it's the gender-critical that don't agree with that, who tend to be on the opposite end of the argument. I'm telling you I'm of male gender, and I'm telling you male gender has meanings and associations that I identity with.
And your response to me thus far has been to insist I don't think it has any meaning (despite what I'm directly telling you), and then to subsequently insist that because it has no meaning-- which I don't agree with-- therefore my idea of self-identity is also meaningless.
Ya see the bloody problem here? You're stubbornly ascribing beliefs to your opponents again.
If someone comes to me and says they are a trans-man, am I to assume that I identify with or understand them in ways that I could not if they were just a woman? Do they think they understand me better than they would if I was a woman? I don't think any of that is true or expected. So then what is that identity signifying?
If someone says to you they're a (cis) man, are you to assume you identify or understand them in ways you couldn't it you were a woman?
These questions apply equally across the board, cis, trans and NB. And yet they only become this odd sticking point when someone wants to whine about trans people.